Module Coq.Bool.Sumbool

Here are collected some results about the type sumbool (see INIT/Specif.v) sumbool A B, which is written {A}+{B}, is the informative disjunction "A or B", where A and B are logical propositions. Its extraction is isomorphic to the type of booleans.

A boolean is either true or false, and this is decidable

Lemma sumbool_of_bool : (b:bool) {b=true}+{b=false}.
Proof.
  Induction b; Auto.
Save.

Hints Resolve sumbool_of_bool : bool.

Lemma bool_eq_rec : (b:bool)(P:bool->Set)
                    ((b=true)->(P true))->((b=false)->(P false))->(P b).
Induction b; Auto.
Save.

Lemma bool_eq_ind : (b:bool)(P:bool->Prop)
                    ((b=true)->(P true))->((b=false)->(P false))->(P b).
Induction b; Auto.
Save.

Logic connectives on type sumbool

Section connectives.

Variables A,B,C,D : Prop.

Hypothesis H1 : {A}+{B}.
Hypothesis H2 : {C}+{D}.

Lemma sumbool_and : {A/\C}+{B\/D}.
Proof.
Case H1; Case H2; Auto.
Save.

Lemma sumbool_or : {A\/C}+{B/\D}.
Proof.
Case H1; Case H2; Auto.
Save.

Lemma sumbool_not : {B}+{A}.
Proof.
Case H1; Auto.
Save.

End connectives.

Hints Resolve sumbool_and sumbool_or sumbool_not : core.


Index